Lifting degeneracy in degenerate perturbation

Question

What is the idea behind finding a set of commuting observables to lift the degeneracy in perturbation theory? I just started a course in quantum mechanics and I do not understand how it works.

My professor was telling us during the lecture that the idea is to diagonalize the Hamiltonian, but I did not fully understand it.

NDewolf · Answer

When a set of observables commutes, they have the same eigenvectors (you can easily check this yourself). Hence if you can find a set of commuting observables, you can find a common set of eigenvectors and these (partially if the set is not complete) diagonalize those observables and hence your Hamiltonian.

Using this technique it is sometimes possible to find a new basis in which the energy eigenvalues are not degenerate anymore.

Lifting degeneracy in degenerate perturbation

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